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NAME

       PCGGQRF  -  compute  a  generalized  QR  factorization  of  an  N-by-M  matrix  sub( A ) =
       A(IA:IA+N-1,JA:JA+M-1) and an N-by-P matrix sub( B ) = B(IB:IB+N-1,JB:JB+P-1)

SYNOPSIS

       SUBROUTINE PCGGQRF( N, M, P, A, IA, JA, DESCA, TAUA, B, IB, JB, DESCB, TAUB, WORK,  LWORK,
                           INFO )

           INTEGER         IA, IB, INFO, JA, JB, LWORK, M, N, P

           INTEGER         DESCA( * ), DESCB( * )

           COMPLEX         A( * ), B( * ), TAUA( * ), TAUB( * ), WORK( * )

PURPOSE

       PCGGQRF  computes  a  generalized  QR  factorization  of  an  N-by-M  matrix  sub(  A  ) =
       A(IA:IA+N-1,JA:JA+M-1) and an N-by-P matrix sub( B ) = B(IB:IB+N-1,JB:JB+P-1):

                   sub( A ) = Q*R,        sub( B ) = Q*T*Z,

       where Q is an N-by-N unitary matrix, Z is a P-by-P unitary matrix, and R and T assume  one
       of the forms:

       if N >= M,  R = ( R11 ) M  ,   or if N < M,  R = ( R11  R12 ) N,
                       (  0  ) N-M                         N   M-N
                          M

       where R11 is upper triangular, and

       if N <= P,  T = ( 0  T12 ) N,   or if N > P,  T = ( T11 ) N-P,
                        P-N  N                           ( T21 ) P
                                                            P

       where T12 or T21 is upper triangular.

       In  particular,  if  sub( B ) is square and nonsingular, the GQR factorization of sub( A )
       and sub( B ) implicitly gives the QR factorization of inv( sub( B ) )* sub( A ):

                    inv( sub( B ) )*sub( A )= Z'*(inv(T)*R)

       where inv( sub( B ) ) denotes the inverse of the matrix sub(  B  ),  and  Z'  denotes  the
       conjugate transpose of matrix Z.

       Notes
       =====

       Each  global  data  object  is described by an associated description vector.  This vector
       stores the information required to establish the mapping between an object element and its
       corresponding process and memory location.

       Let  A be a generic term for any 2D block cyclicly distributed array.  Such a global array
       has an associated description vector DESCA.  In the following comments,  the  character  _
       should be read as "of the global array".

       NOTATION        STORED IN      EXPLANATION
       ---------------   --------------   --------------------------------------  DTYPE_A(global)
       DESCA( DTYPE_ )The descriptor type.  In this case,
                                      DTYPE_A = 1.
       CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating
                                      the BLACS process grid A is distribu-
                                      ted over. The context itself is glo-
                                      bal, but the handle (the integer
                                      value) may vary.
       M_A    (global) DESCA( M_ )    The number of rows in the global
                                      array A.
       N_A    (global) DESCA( N_ )    The number of columns in the global
                                      array A.
       MB_A   (global) DESCA( MB_ )   The blocking factor used to distribute
                                      the rows of the array.
       NB_A   (global) DESCA( NB_ )   The blocking factor used to distribute
                                      the columns of the array.
       RSRC_A (global) DESCA( RSRC_ ) The process row over which the first
                                      row of the array A is distributed.  CSRC_A (global)  DESCA(
       CSRC_ ) The process column over which the
                                      first column of the array A is
                                      distributed.
       LLD_A  (local)  DESCA( LLD_ )  The leading dimension of the local
                                      array.  LLD_A >= MAX(1,LOCr(M_A)).

       Let  K  be  the  number  of  rows  or columns of a distributed matrix, and assume that its
       process grid has dimension p x q.
       LOCr( K ) denotes the number of elements of K that a  process  would  receive  if  K  were
       distributed over the p processes of its process column.
       Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K
       were distributed over the q processes of its process row.
       The values of LOCr() and LOCc() may be  determined  via  a  call  to  the  ScaLAPACK  tool
       function, NUMROC:
               LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ),
               LOCc(  N  )  =  NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ).  An upper bound for these
       quantities may be computed by:
               LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A
               LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A

ARGUMENTS

       N       (global input) INTEGER
               The number of rows to be operated on i.e the number of  rows  of  the  distributed
               submatrices sub( A ) and sub( B ). N >= 0.

       M       (global input) INTEGER
               The  number  of  columns  to  be  operated  on  i.e  the  number of columns of the
               distributed submatrix sub( A ).  M >= 0.

       P       (global input) INTEGER
               The number of columns to  be  operated  on  i.e  the  number  of  columns  of  the
               distributed submatrix sub( B ).  P >= 0.

       A       (local input/local output) COMPLEX pointer into the
               local  memory to an array of dimension (LLD_A, LOCc(JA+M-1)).  On entry, the local
               pieces of the N-by-M distributed matrix sub( A ) which  is  to  be  factored.   On
               exit, the elements on and above the diagonal of sub( A ) contain the min(N,M) by M
               upper trapezoidal matrix R (R is upper triangular if N >= M); the  elements  below
               the  diagonal, with the array TAUA, represent the unitary matrix Q as a product of
               min(N,M) elementary reflectors (see  Further  Details).   IA       (global  input)
               INTEGER The row index in the global array A indicating the first row of sub( A ).

       JA      (global input) INTEGER
               The column index in the global array A indicating the first column of sub( A ).

       DESCA   (global and local input) INTEGER array of dimension DLEN_.
               The array descriptor for the distributed matrix A.

       TAUA    (local output) COMPLEX, array, dimension
               LOCc(JA+MIN(N,M)-1). This array contains the scalar factors TAUA of the elementary
               reflectors which represent the unitary matrix Q. TAUA is tied to  the  distributed
               matrix  A.  (see  Further  Details).   B        (local input/local output) COMPLEX
               pointer into the local memory to an array of dimension (LLD_B, LOCc(JB+P-1)).   On
               entry,  the  local pieces of the N-by-P distributed matrix sub( B ) which is to be
               factored. On exit, if N <= P, the  upper  triangle  of  B(IB:IB+N-1,JB+P-N:JB+P-1)
               contains the N by N upper triangular matrix T; if N > P, the elements on and above
               the (N-P)-th subdiagonal contain the N  by  P  upper  trapezoidal  matrix  T;  the
               remaining  elements,  with  the  array  TAUB,  represent the unitary matrix Z as a
               product of elementary reflectors (see Further Details).   IB       (global  input)
               INTEGER The row index in the global array B indicating the first row of sub( B ).

       JB      (global input) INTEGER
               The column index in the global array B indicating the first column of sub( B ).

       DESCB   (global and local input) INTEGER array of dimension DLEN_.
               The array descriptor for the distributed matrix B.

       TAUB    (local output) COMPLEX, array, dimension LOCr(IB+N-1)
               This  array  contains  the  scalar  factors  of  the  elementary  reflectors which
               represent the unitary matrix Z. TAUB is tied to  the  distributed  matrix  B  (see
               Further Details).  WORK    (local workspace/local output) COMPLEX array, dimension
               (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK.

       LWORK   (local or global input) INTEGER
               The dimension of the array WORK.  LWORK is local input and must be at least  LWORK
               >=  MAX( NB_A * ( NpA0 + MqA0 + NB_A ), MAX( (NB_A*(NB_A-1))/2, (PqB0 + NpB0)*NB_A
               ) + NB_A * NB_A, MB_B * ( NpB0 + PqB0 + MB_B ) ), where

               IROFFA = MOD( IA-1, MB_A ), ICOFFA = MOD( JA-1, NB_A  ),  IAROW   =  INDXG2P(  IA,
               MB_A,  MYROW, RSRC_A, NPROW ), IACOL  = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ),
               NpA0   = NUMROC( N+IROFFA, MB_A, MYROW, IAROW, NPROW ), MqA0   = NUMROC( M+ICOFFA,
               NB_A, MYCOL, IACOL, NPCOL ),

               IROFFB  =  MOD(  IB-1,  MB_B  ), ICOFFB = MOD( JB-1, NB_B ), IBROW  = INDXG2P( IB,
               MB_B, MYROW, RSRC_B, NPROW ), IBCOL  = INDXG2P( JB, NB_B, MYCOL, CSRC_B, NPCOL  ),
               NpB0   = NUMROC( N+IROFFB, MB_B, MYROW, IBROW, NPROW ), PqB0   = NUMROC( P+ICOFFB,
               NB_B, MYCOL, IBCOL, NPCOL ),

               and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL,  NPROW  and  NPCOL
               can be determined by calling the subroutine BLACS_GRIDINFO.

               If  LWORK  =  -1, then LWORK is global input and a workspace query is assumed; the
               routine only calculates the minimum and optimal size for all work arrays. Each  of
               these  values  is returned in the first entry of the corresponding work array, and
               no error message is issued by PXERBLA.

       INFO    (global output) INTEGER
               = 0:  successful exit
               < 0:  If the i-th argument is an array and the j-entry had an illegal value,  then
               INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then
               INFO = -i.

FURTHER DETAILS

       The matrix Q is represented as a product of elementary reflectors

          Q = H(ja) H(ja+1) . . . H(ja+k-1), where k = min(n,m).

       Each H(i) has the form

          H(i) = I - taua * v * v'

       where taua is a complex scalar, and v is a complex vector with v(1:i-1) = 0 and v(i) =  1;
       v(i+1:n) is stored on exit in
       A(ia+i:ia+n-1,ja+i-1), and taua in TAUA(ja+i-1).
       To form Q explicitly, use ScaLAPACK subroutine PCUNGQR.
       To use Q to update another matrix, use ScaLAPACK subroutine PCUNMQR.

       The matrix Z is represented as a product of elementary reflectors

          Z = H(ib)' H(ib+1)' . . . H(ib+k-1)', where k = min(n,p).

       Each H(i) has the form

          H(i) = I - taub * v * v'

       where  taub  is a complex scalar, and v is a complex vector with v(p-k+i+1:p) = 0 and v(p-
       k+i) = 1; conjg(v(1:p-k+i-1)) is stored on exit in B(ib+n-k+i-1,jb:jb+p-k+i-2),  and  taub
       in TAUB(ib+n-k+i-1).  To form Z explicitly, use ScaLAPACK subroutine PCUNGRQ.
       To use Z to update another matrix, use ScaLAPACK subroutine PCUNMRQ.

       Alignment requirements
       ======================

       The  distributed  submatrices sub( A ) and sub( B ) must verify some alignment properties,
       namely the following expression should be true:

       ( MB_A.EQ.MB_B .AND. IROFFA.EQ.IROFFB .AND. IAROW.EQ.IBROW )